The first of these two figures shows the index plot of the pH levels for both samples. The second figure shows the sample histograms of these samples, which are clearly not Normal-like. Therefore, the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |Independent T-Test]] would not be appropriate to analyze these data.

+

The first figure shows the index plot of the pH levels for both samples. The second figure shows the sample histograms of these samples, which are clearly not Normal-like. Therefore, the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |Independent T-Test]] would not be appropriate to analyze these data.

Intuitively, we may consider these group differences significantly large, especially if we look at the [[SOCR_EduMaterials_Activities_BoxPlot | Box-and-Whisker Plots]], but this is a qualitative inference that demands a more quantitative statistical analysis that can back up our intuition.

Intuitively, we may consider these group differences significantly large, especially if we look at the [[SOCR_EduMaterials_Activities_BoxPlot | Box-and-Whisker Plots]], but this is a qualitative inference that demands a more quantitative statistical analysis that can back up our intuition.

Clearly the p-value < 0.05, and therefore our data provides sufficient evidence to reject the null hypothesis and therefore assumes that there were significant pH differences between the two soil lots tested in this experiment..

+

Clearly the p-value < 0.05, and therefore our data provides sufficient evidence to reject the null hypothesis. So we assumes that there were significant pH differences between the two soil lots tested in this experiment.

: One-Sided P-Value for Sample2 < Sample1: 0.00040

: One-Sided P-Value for Sample2 < Sample1: 0.00040

Line 93:

Line 93:

Both types of tests answer the same question, but treat data differently.

Both types of tests answer the same question, but treat data differently.

*The WMW test uses rank ordering

*The WMW test uses rank ordering

-

: Positive: doesn’t depend on normality or population parameters

+

: Positive: Doesn’t depend on normality or population parameters

-

: Negative: distribution free lacks power because it doesn't use all the info in the data

+

: Negative: Distribution free lacks power because it doesn't use all the info in the data

* The T-test uses the raw measurements

* The T-test uses the raw measurements

-

: Positive : Incorporates all of the data into calculations

+

: Positive: Incorporates all of the data into calculations

-

: Negative : Must meet normality assumption

+

: Negative: Must meet normality assumption

* Neither test is uniformly superior. If the data are normally distributed we use the T-test. If the data are not normal use the WMW test.

* Neither test is uniformly superior. If the data are normally distributed we use the T-test. If the data are not normal use the WMW test.

Motivational Example

Nine observations of surface soil pH were made at two different (independent) locations. Does the data suggest that the true mean soil pH values differs for the two locations? Note that there is no pairing in this design, even though this is a balanced design with 9 observations in each (independent) group. Test using α = 0.05, and be sure to check any necessary assumptions for the validity of your test.

Location 1

Location 2

8.10

7.85

7.89

7.30

8.00

7.73

7.85

7.27

8.01

7.58

7.82

7.27

7.99

7.50

7.80

7.23

7.93

7.41

We see the clear analogy of this study design to the independent 2-sample designs we saw before. However, if we were to plot these data we can see that their distributions may be different or not even symmetric, unimodal and bell-shaped (i.e., not Normal). Therefore, we cannot use the Independent T-Test to test a Null-hypothesis that the centers of the two distributions (that the 2 samples came from) are identical, using this parametric test.

The first figure shows the index plot of the pH levels for both samples. The second figure shows the sample histograms of these samples, which are clearly not Normal-like. Therefore, the Independent T-Test would not be appropriate to analyze these data.

Intuitively, we may consider these group differences significantly large, especially if we look at the Box-and-Whisker Plots, but this is a qualitative inference that demands a more quantitative statistical analysis that can back up our intuition.

The Wilcoxon-Mann-Whitney Test

The Wilcoxon-Mann-Whitney (WMW) Test (also known as Mann-Whitney U Test, Mann-Whitney-Wilcoxon Test, or Wilcoxon Rank-Sum Test) is a non-parametric test for assessing whether two samples come from the same distribution. The null hypothesis is that the two samples are drawn from a single population, and therefore that their probability distributions are equal. It requires that the two samples are independent, and that the observations are ordinal or continuous measurements.

Calculations

The U statistic for the WMW test may be approximated for sample sizes above about 20 using the Normal Distribution.

Clearly the p-value < 0.05, and therefore our data provides sufficient evidence to reject the null hypothesis. So we assumes that there were significant pH differences between the two soil lots tested in this experiment.

Both types of tests answer the same question, but treat data differently.

The WMW test uses rank ordering

Positive: Doesn’t depend on normality or population parameters

Negative: Distribution free lacks power because it doesn't use all the info in the data

The T-test uses the raw measurements

Positive: Incorporates all of the data into calculations

Negative: Must meet normality assumption

Neither test is uniformly superior. If the data are normally distributed we use the T-test. If the data are not normal use the WMW test.

Practice Examples

Urinary Fluoride Concentration in Cattle

The urinary fluoride concentration (ppm) was measured both for a sample of livestock grazing in an area previously exposed to fluoride pollution and also for a similar sample of livestock grazing in an unpolluted area.